Option pricing with Levy Process

Option pricing with Levy Process

Eric Benhamou

First Version: April 2000, This version: July 11, 2000. JEL Classi&cation: G13

MSC classi&cation: 62P05

Keywords: Levy process, Fourier and Laplace transform, Smile.

Abstract

In this paper, we assume that log returns can be modelled by a Levy process. We give explicit formulae for option prices by means of the Fourier transform. We explain how to infer the characteristics of the Levy process from option prices. This enables us to generate an implicit volatility surface implied by market data. This model is of particular interest since it extends the seminal Black Scholes [1973] model consistently with volatility smile.

1

Introduction

It is now widely accepted that markets differ from the seminal Black Scholes [1973] model. The empirical literature has extensively reported on these abnormalities, espe-cially on two of them, which indeed are closely linked. First, it is has been shown that unconditional return show excess kurtosis and skewness, inconsistent with normality assumptions (see Mandelbrot [1963] and Fama [1965] for the former ones, Kon [1984] and Jorion [1988] for more recent works, Bates [1996] for more references). Second, research has concentrated its attention on the implied volatility smile or skew (see Dumas et al. [1995] for a survey). Interestingly, the second fact is just another hint of

∗_{Financial Markets Group, London School of Economics, and Centre de Mathématiques} Ap-pliquées, Ecole Polytechnique. Address: FMG, Room G303, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom, Tel 0171-955-7895, Fax 0171-242-1006. E-mail: [email protected]

1 INTRODUCTION 2

the non-normality of returns. However, research has focussed at implied Black Scholes volatility since implied volatility has become a key concept for option pricing. Option prices are often quoted by their implied volatility. A more rigorous justi&cation is the less volatile character as well as the better predictability of volatility compared to prices.

Research has extensively offered methods to cope with the smile effect. Classi-cally, these attempts can be divided into two different families: parametric and non parametric ones.

In the parametric methods, the equation of the evolution of the underlying process is speci&ed as a particular functional form. This description can consist either in a continuous diffusion process with a so called deterministic volatility (Rubinstein [1994], Dupire [1993] and Derman and Kani [1994]) or with a stochastic volatility process (Hull and White [1987], Wiggings [1987], Melino and Turnbull [1990], Stein and Stein [1991], Amin and Ng [1993] and Heston [1992]) or in a model with jumps (Aase [1993], Ahn and Thompson [1988], Amin [1993], Bates [1991] , Jarrow [1984], Merton [1976]).

Other works close in spirit are assuming constant elasticity of volatility distribution often called power-law(Cox Ross [1976]) or a mapping principle between normal and lognormal distributions (Hagan [1998], Pradier and Lewicki [1999]).

The second type of methods is the inference of the underlying distribution from market data. This has been called the expansion methods where one infers the different terms of the expansion and can reconstitute the distribution (Jarrow and Rud [1982], Bouchaud et al.[1998], Abken et al. [1996]).

The motivation of this paper is to present a semi-parametric method for modelling the smile effect. We assume that the underlying price process can be modelled as Levy process. However, we give no speci&c conditions on the underlying price process except some technical conditions. Since Levy processes include continuous time diffusion as well as jump process, this approach encompasses many of the previous method. It extends the Black Scholes model to any type of Levy process for the underlying.

The remainder of this paper is organized as follows. In section 2, we introduce some characteristic of Levy processes, its Laplace and Fourier transform. Section 3 explains how to compute option prices. Section 4 examines the volatility smile issue. We conclude brie! y giving further developments.

2 LEVY PROCESS AND PROPERTIES 3

2

Levy process and properties

2.1

Modelling assumptions

We consider a continuous time trading economy with in&nite horizon. The uncertainty in this economy is classically modeled by a complete probability space(Ω, F,_Q).The information evolves according to the augmented &ltration _{Ft, t∈R+} generated by

a standard Brownian motion (Wt)t∈R+. We assume that there exits a Levy process

(Xt)t∈R+. We de&ne a Levy process as a stochastic process, adapted to the Brownian

&ltration _{Ft, t∈R+}, which satis&es the following property: (Xt)_t∈R+ is with

inde-pendent stationary increments and is centered in its origin. X0 = 0 almost surely.

This Levy process is not assumed to be a stable Levy process, with stable Levy law. This process does not necessary satisfy a scaling law. We assume as well that the Laplace transform of the Levy process is bounded. There exits τ >0,λu >0 so that

for every λd∈]− ∞,λu], t∈[0,τ], the Laplace transform λ7−→E £

eλXt¤_{is bounded}

by two strictly positive constant over [λd,λu]. There exist Bd > 0, Bu > 0 so that

∀λ_∈[λd,λu],∀t∈[0,τ]

Bd≤E h

eλXti

≤Bu (1)

We assume that the underlying (St)_t∈R+ can be modelled as a continuous time

process, written as a function of a geometric Brownian motion times the exponential of a Levy process with no Brownian part. This leads to the following decomposition

St=S0e ³ r−σ22 ´ T+σWT+Xt (2) It is worth noticing that the condition on the Laplace transform of the Levy process for negative values of λ implies that the event that the underlying equals zero, is of nul measure: P(St= 0) = 0for everyt∈R+. It is worth noticing as well that we take

the de&nition of an extended Laplace transform since we allow for both positive and negative values for λ as opposed to the traditional Laplace only de&ned for positive values ofλ.We can then introduce two characterizations of the Levy process. The &rst one is based on the Laplace transform, whereas the second on the Fourier transform

2 LEVY PROCESS AND PROPERTIES 4

2.2

First Characterization of the Levy process

2.2.1 Exponent for the Laplace transform

Proposition 1 There exists a functionφ:]_−∞,λu]→R de&ned as for everyt∈R+,

λ_∈]_{− ∞},λu]

E h

eλXti_=etφ(λ)

This function is called the Levy-Laplace exponent.

Proof: Let take λ_∈]_{− ∞},λu]. We can &rst show that for every t∈R+,E £

eλXt¤

exits and is &nite £eλXt¤_{<+∞.For everyn∈N}∗_,t∈R+_, EheλXnti ₌ Eheλ(Xnt−X(n−1)t)_eλXt(n−1) i = _E h eλ(Xnt−X(n−1)t) i E h eλXt(n−1) i = _EheλXti EheλXt(n−1) i

where we have used in the last two equation &rst the independence between increments and second the stationarity of increments. This leads to

EheλXti₌

EheλXt/n

in

Fornso thatt/n <τ, this proves that the above quantity exists and is bounded. The independence and stationarity of increments leads as well that for every u _{∈ R}+and v_∈R+,

EheλXui

EheλXvi₌

EheλXu+vi

This indicates that the function fλ:t7→E £

eλXt¤ _satis&es

fλ(u+v) =fλ(u).fλ(v) (3)

This function is as well continuous in zero. First, it is easy to see that this function should satisfy fλ(0) = 0 or 1. Because of assumption (1), the only possible case is fλ(0) =1. Second, if the function were not continuous in zero, it would imply that there would exist a sequence (εn)n∈N of real numbers strictly decreasing to zero so

that lim ε↓0E

£

e−λXε¤_{is not equal to 1. This would mean that there existη>0so that} ¯ ¯ ¯E h eλXε i −1 ¯ ¯ ¯>η

2 LEVY PROCESS AND PROPERTIES 5

Denoting by (qn)n∈N a sequence of rational numbers so that εnqn ∈ £_τ 2,τ ¤ ,we have qn → n→+∞+∞leading to EheλXqnεni₌³ EheλXεn i´qn

We have then two cases: either_E£eλXεn¤_{≥1+ηfor an in&nity of terms and the value}

+_∞is an accumulation point of the sequence(Xεn)n∈Nor it is the case ofE

£

eλXεn¤≤

1₋η to be satis&ed by an in&nity of terms and the value0 is an accumulation point of the sequence (Xεn)n∈N. Both of cases contradict the original assumption (1). We

have proved that the function fλ : t 7→ E £

eλXt¤ _{is an automorphism from (R}+_,+)

to (R+_{,∗) which is continuous in zero. It can therefore be written as an exponential}

function

fλ(t) =etφ(λ)

The preceding arguments assume only λ_∈]_{− ∞},λu]. As a conclusion, we have built

a function φ:t_7→φ(λ).¤

Remark 1 It is straightforward to get the different momentum of the underlying pro-cess EhS_tλi=S₀λet ³ λ³µ−σ22 ´ +σ2₂λ2+φ(λ)´ (4) 2.2.2 Particular Cases

Let us de&ne (Wt)t∈R+ as a Brownian motion,(Nt)t∈R+ a Poisson process of intensity

θand a sequence of independent variable independent identically distributed(Uj)_j∈N∗

with value in ]₋1,+_∞[and U0=1almost surely. The common law of the variables (Uj)_j∈N∗ is denoted byL, associated with a variableU. Foru ∈]− ∞,1],we assume

that _E[(1+U)u] is &nite. Let us assume that the respective &ltrations spanned by

(Wt)_t∈R+, (Nt)_t∈R+ and (Uj)_j∈N∗ are independent. Let us denote by (Ft)_t∈R+ the

&ltration spanned by the stochastic variables (Ws)s≤t, (Ns)s≤t and (Uj)_j≤Nt. In this

framework, (Wt)_t∈R+ is still a Brownian motion adapted to the &ltration (Ft)_t∈R+

respectively (Nt)t∈R+ a Poisson process of intensity θ. The jump time of the

Pois-son process denoted by (τj)_j∈N∗ are still stopping time for the &ltration (Ft)t∈R+.

We assume that the underlying security can be modelled as a risky asset with some stochastic jumps of stochastic intensity(Uj)_j∈N∗which occur according to the Poisson

2 LEVY PROCESS AND PROPERTIES 6

process(Nt)t∈R+.Between two jumps, the risky asset can be modelled by a standard

geometric Brownian motion, as in the Black Scholes model [1973] with a determin-ist drift µ and a volatility σ. This leads to the following formula for the underlying security St=e ³ µ−σ22 ´ t+σWt Nt Y j=1 (1+Uj) (5)

with the convention thatQ0_j=1(1+Uj) =1. This can be seen as a particular example

of our example since in this case the Levy process is

Xt= ln   Nt Y j=1 (1+Uj)  

We have the following proposition so as to calculate the Levy exponentφ:]_−∞,λu]→ R. Proposition 2 E h Sλ_t i =et ³ λ³µ−σ2 2 ´ +σ2λ2 2 +θ(E(1+U) λ −1)´ Proof: EhS_tλi = _E  eλ ³ µ−σ22 ´ t+λσWt Nt Y j=1 (1+Uj)λ   = eλ ³ µ−σ22 ´ t+(λσ₂)2t E   Nt Y j=1 (1+Uj)λ  

The last equation holds because of the independence of the stochastic variables(Ws)s≤t,

(Ns)s≤tand (Uj)_j≤Nt. We have then

E   Nt Y j=1 (1+Uj)λ   = +∞ X n=0 E   NYt=n j=1 (1+Uj)λ ¯ ¯ ¯ ¯ ¯ ¯Nt=n  _P(Nt=n) = +_X∞θ n=0 ³ E³(1+U)λ´´ne−θt(θt) n n! = exp ³ θt ³ E ³ (1+U)λ ´ −1 ´´ ¤

The proposition 2 leads to the following Levy exponent function: φ(λ) =θ ³ E ³ (1+U)λ ´ −1 ´

2 LEVY PROCESS AND PROPERTIES 7

2.3

Second Characterization: Characteristic function

Another characteristic of the Levy process is its characteristic function which can be analyzed as its Fourier Transform. This has the advantage to be very ! exible since the Fourier transform _E£eiλX¤is always de&ned and no condition is required on the momentum. It is also numerically very efficient by means of Fast Fourier transform algorithms.

2.3.1 Fourier transform and characteristic function

Let us remind some preliminary results. We assume that Xt is a Levy process. We

have the following proposition:

Proposition 3 There exists a functionψ:]_{− ∞},+_∞[_→R so that for every t_∈R+, for every λ_∈R

EheiλXt

i

=etψ(λ)

with ψ(λ) =iµλ₋σ₂2λ2₋R_R¡1₋eiλx_+iλx1

{|x|<1}Π(dx)

¢

with µ_∈Rand Π, called the Levy measure, is a positive measure on _R so that R_RM in¡1, x2¢Π(dx)<_∞.The term σ₂2λ2 is called the Brownian part of the process. The parameter µ is the drift of the Levy process.

Proof:See Bertoin [1997].¤

The function ψ :]_{− ∞},+_∞[_{→ R} is called the Levy-Khintchine exponent. To estimate the Brownian part, we use the following proposition:

Proposition 4 There exists a limit to the following ratio:

lim |λ|→∞ ψ(λ) λ2 =− σ2 2 Proof:We &rst notice that

lim |λ|→∞

1₋eiλx+iλx1_{|x|<1}

λ2 = 0

It can be shown (see Bertoin [1997]) for every x_∈R

3 OPTION PRICE 8

as well as for every x_∈R, for everyλ_∈R ¯

¯₋sin (λx) +λx1_{|x|<1}

¯

¯_≤M in³1+_|λ_|,(λx)2´

Combining the different results, we get that for _|λ_|≥2

¯

¯1₋eiλx+iλx1_{|x|<1}¯¯

λ2 ≤3M in

¡

1, x2¢ We can conclude by dominated convergence that

lim |λ|→∞ 1 λ2 Z R ³ 1₋eiλx+iλx1_{|x|<1}Π(dx) ´ = 0 ¤ 2.3.2 Particular case

In the case of the process given in the section 2.2.2 by equation (5), we can calculate explicitly the characteristic function. This is summarized by the following proposition

Proposition 5 For everyt_∈R, and every λ_∈R

E h eiλXt i =et ³ iλ³µ−σ2 2 ´ −σ2λ2 2 +θ(E(1+U) iλ −1)´ Proof: same as in proposition 2.¤

3

Option price

The interest of this extension of the Black Scholes model is its tractability. We can &nd explicit formula for the price of vanilla option as shown in the following subsection. To get a price, we assume that the discounted asset is a martingale under the natural probability measure _Q of our probability space (Ω, F,_Q). We impose this restriction so as to be have a unique martingale-measure used for pricing purposes. It can be shown that there exists an in&nity of equivalent martingale measures under which ¡

e−rTST ¢

T∈Ris a martingale. The correct price is obtained as the expected discounted

payoff under this measure. The price of the call option of strike K, maturity T, with an initial underlying level of S0 calculated as the expectation of the discounted

payoff e−rT(ST −K)+ under one of this equivalent martingale measure is dense in

the interval ofh¡S0−e−rTK

¢+

, S0

i

3 OPTION PRICE 9

3.1

Laplace transform and option price

¢

is a martingale under _P. This leads to the following constraints:

Proposition 6

µ=r₋φ(1) (6)

Proof: ¡e−rtSt ¢

t∈R+ is a martingale. This implies that

E£e−rtSt ¤

=S0

or using the proposition 2 this leads to et ³ −r+³µ−σ2 2 ´ +σ2 2 +φ(1) ´ =1 or the condition (6).¤ 3.1.1 General formula

Let us denote by PCall respectively PP ut the price of a call option, respectively the price of a put option, and by P_BSCall(S0, K, T, r,σ) (respectively PBSP ut(S0, K, T, r,σ))

the price of a Black Scholes call (respectively put) with an initial underlying level of S0 a strike of K, a maturity ofT, a risk free rate of r and a volatility ofσ. Similarly

to the formula given by Hull and White [1987] for stochastic volatility, the price of the option is given by the following proposition

Proposition 7 The price of a vanilla option is given by

Pi =_EhP_BSi ³S0e−Tφ(1)eXT, K, T, r,σ

´i

(7)

where istands for either call or put.

Proof: Since a put option onST with strike can be seen as a call option on(−ST)

with a strike ₋K (K₋ST)+ = (−ST −(−K))+, we only examine the case of the

call option. The price of the option is calculated as the expectation of the discounted payoff E£e−rT(ST −K)+ ¤ = _E " e−rT µ S0e ³ µ−σ22 ´ T+σWt eXT _−K ¶+# = _E " e−rT µ³S0e−Tφ(1) ´ e ³ r−σ22 ´ T+σWt eXT _−K ¶+#

3 OPTION PRICE 10

where we have used the no-arbitrage condition on the drift term (6).Using conditional expectation, we get E£e−rT (ST −K)+ ¤ = _E " E " e−rT µ³S0e−Tφ(1)eXT ´ e ³ r−σ22 ´ T+σWt −K ¶+¯¯ ¯ ¯ ¯XT ##

Where the conditional expectation can be interpreted in the Black Scholes model as a closed formula, leading to the &nal result.¤

The same methodology can be applied to binary and range option with payoff equal to

f(x) =1_{ex_>K}

for a binary option and

f(x) = (ex₋K1)1_{K1≤ex<K2}

Denoting byP_BsBin respectivelyP_BSRangethe Black Scholes price of a binary respectively a range option, we have the following proposition

Proposition 8 The price of a binary respectively a range option is given by

Pi =_E h P_BSi ³ S0e−Tφ(1)eXT, K, T, r,σ ´i (8)

where istands for either Bin or Range.

Proof: same as above.¤

3.1.2 Particular case

In the case of the process given in the section 2.2.2 by equation (5) this leads to the following results

Proposition 9 The no arbitrage condition (6) leads to

µ=r₋θ_E(U) (9)

and the price of a vanilla option is given by

Pi =_E  P_BSi  S0e−TθE(U) NT Y j=1 (1+Uj), K, T, r,σ     (10)

3 OPTION PRICE 11

Let us now assume that the intensity process of the jumps modelled by the variable

(1+U) follows a lognormal distribution with meanmand volatilityv2. We then get an explicit formula for the vanilla option price, as stated by the following proposition, similar to the Merton s formula [1976]:

Proposition 10 Pi =e−θ(1+c)T +∞ X i=1 (θ(1+k)T)n n! P i BS(S0, K, T, rn,σn) (11) with c = _E[U] =em+v2/2₋1 rn = r−θc+ n¡m+v2_/2¢ T σn = µ σ2+nv 2 T ¶1/2

with i=c or p corresponding to a call or put option.

Proof: Introducing the variable ε = 1 for a call option and ε = ₋1 for a put option, we treat the two options in the same way. SinceU+1follows a lognormal law with meanmand volatility v2, the expectation ofU is given by_E[U] =em+v2/2₋1. Denoting byS0e−T cθ and using the pricing formula (10), we can write the price of the

option Pi = +∞ X i=0 E  P_BSi  S0e−T cθ n Y j=1 (1+Uj), K, T, r,σ    e−θT(θT) n n! (12)

The random variables (1+Ui)i∈N∗ are independently distributed, with a lognormal

distribution with mean m and volatility v2_{. Their product} Qn

j=1(1+Uj) follows a

lognormal distribution with mean nm and volatility nv2. Denoting by g a centered normalized normal distribution (g_∼N(0,1)), byS=S0e−T cθ, the equation (12) can

be rewritten as Pi = +∞ X i=0 E       εSenm+√nvgN Ã ε Ã ln ³ S e−rT K ´ +nm+√nvg σ√T + σ 2 √ T !! −εKe−rT_N Ã ε Ã ln³ S e−rT K ´ +nm+√nvg σ√T − σ 2 √ T !!      e −θT(θT) n n! (13)

It can be shown that for g_∼N(0,1) and for everya, b, c_∈R,

E[eagN(bg+c)] =ea2/2N µ c+ab √ 1+b2 ¶ (14)

3 OPTION PRICE 12

Using the equation (14), we get

E     ³ εSenm+√nvg´ N Ã ε Ã ln³ S e−rT K ´ +nm+√nvg σ√T + σ 2 √ T !!     = εSenm+nv 2 2 N  εln ¡ _S e−rT_K ¢ +nm+ σ₂2T +nv2 σ√T q 1+_σnv2_T2   and E " εKe−rTN Ã ε Ã ln¡_e−rTS_K ¢ +nm+√nvg σ√T − σ 2 √ T !!# = εKe−rTN  εln ¡ _S e−rT_K ¢ +nm₋ σ₂2T σ√T q 1+_σnv2_T2  

which can be rewritten as

Pi= +∞ X i=0                    εS0e à nm+nv2 2 T −θc ! T N     ε ln³S0 K ´ +  n µ m+v2 2 ¶ T +r−θc  T+1₂³σ2+nv_T2´T ³ σ2₊nv2 T ´1/2_√ T      −εKe−rTN     ε ln³S0 K ´ +  n µ m+v2 2 ¶ T +r−θc  T−12 ³ σ2₊nv2 T ´ T ³ σ2₊nv2 T ´1/2_√ T                         e−θT(θT) n n!

which leads to the result (11).¤

Corollary 11 The price of a binary or a range option is given by

Pi =e−θ(1+c)T +∞ X i=1 (θ(1+k)T)n n! P i BS(S0, K, T, rn,σn) (15) with c = _E[U] =em+v2/2₋1 rn = r−θc+ n¡m+v2/2¢ T σn = µ σ2+nv 2 T ¶1/2

with i=Bin or Range according to the option type

3 OPTION PRICE 13

3.2

Fourier transform and convolution product

3.2.1 Preliminary results

We denote by_L1(_R)the linear space of integrable function de&ned on_R, and byF(f)

the Fourier transform of the function f. It is interesting to see that we can interpret any expectation as the convolution product of a function with the density function of our stochastic variable. Let us assume that our underlying security can be written as the exponential of a Levy process with some Brownian part and drift term

St=eXt

Let dpXT (x) be the probability measure of the process XT. Any option price which

can be written as the expectation of some discounted payoffcan be reinterpreted as a convolution product as stated by the following proposition

Proposition 12 Let f be a function f : _{R → R}, continuous, bounded, element of

L1_{(R) so thatF(f) belongs to L}1_{(R). We have}

E[f(Xt)] = ³

F−1³Ffe(.)eTψ(.)´´(0) (16)

withψthe Fourier exponent as de&ned in proposition (3) andFfe(.)the Fourier trans-form of the function fe:fe(x) =f(₋x)

Proof: Letg denote the convolution product of fewithdpXT

g(y) =

Z +∞ −∞

e

f(y₋x)dpXT(x)

g is well de&ned sincef and feis bounded. g is integrable since Z +∞ −∞ | g(y)_|dy _≤ Z +∞ y=−∞ Z +∞ x=−∞ ¯ ¯ ¯ ef(y₋x)_¯¯¯dpXT(x)dy ≤ Z +∞ x=−∞ µZ +∞ y=−∞ ¯ ¯ ¯ ef(y₋x) ¯ ¯ ¯dy ¶ dpXT(x) ≤ ° ° ° ef ° ° ° L1_(R) Z +∞ x=−∞ dpXT (x) ≤ kf_kL1_(R)

3 OPTION PRICE 14

The expectation off(Xt)can be seen as the value in zero of this convolution product

since E[f(Xt)] = Z +∞ −∞ f(x)dpXT(x) = Z +∞ −∞ e f(₋x)dpXT (x) = g(0)

The efficient way of calculating convolution product (see Benhamou [2000] is to use the property of Fourier transform. The Fourier transform of a convolution product is simply the product of the Fourier transform. This leads to multiply the two Fourier transform and invert the Fourier transform of the &nal result. We verify that g(y)

is a continuous function since by assumption f is continuous and bounded. We have veri&ed as well that gbelongs to_L1(_R). Using the fact that the Fourier transform of a convolution product is simply the product of the Fourier transform and that F(f)

belongs to _L1(_R) as well as the characteristic function of the Levy process F(pXT),

we get the F(g) belongs as well to_L1(_R). The validity of the Fourier transform and its inversion is then given by the lemma below.¤

Lemma 13 If f belongs to _L1(_R) is a continuous function so that F(f) belongs to

L1(_R), thenF−1(F(f)) =f

Proof: standard in the Fourier theory (see Bracewell [1965].¤

We cannot apply the result of proposition 12 equation (16) straightforward to the call or put option. This is due to the fact that the payofffunction equal to(ex₋K)+

for a call or (K₋ex)+ does not belong to _L1(_R). However, it is possible to &nd a solution to this problem.

3.2.2 General formula

One solution is to use a truncated version of the payofffunction. LetMc, Mp be two

real numbers satisfying

Mc>lnK > Mp (17)

Let α be a real number so that α >1 Let us de&ne the standard payoff of a vanilla option

3 OPTION PRICE 15

for a call option and

fp(x) = (K₋ex)+

the truncated payoffde&ned as

f_Mc_c(x) = (ex₋K)+³1_{x≤Mc}+e

−α(x−Mc)₁

{x>Mc}

´

for a call option and f_Mp p(x) = (K−e x₎+³₁ {x≥Mp}+e −α(x−Mp)₁ {x<Mp} ´

Proposition 14 These two functions are continuous, with positive values. They be-longs as well to _L1_{(R) and are bounded. They satisfy as well that their Fourier}

transform belongs to _L1(_R).

Proof: We examine the case of the call truncated payoff. The proof goes along the same line for the second function. The function f_Mc_c(x) is continuous as the product of continuous functions. It is positive as the product of two positive functions. It is integrable since for values of x smaller than lnK, it is equal to zero and for large values of x it is equivalent to the function e−(α−1)xeαMp_{. Moreover, this implies that}

this function is bounded since it is a continuous function with asymptotic limits equal to zero. Its Fourier transform can be written as

Z +∞ −∞ f_Mc_c(x)eiλxdx = Z Mc lnK (ex₋K)eiλxdx+ Z +∞ Mc (ex₋K)e−α(x−Mc)_eiλx_dx = ₋Ke iλlnK_(λ+i) ¡ 1+λ2¢λ −Kαe iλMc λ−iα λ¡α2_+λ2¢ +αe(1+iλ)Mcλ 2_{+ (α} −1) +iλ(2₋α) ¡ 1+λ2¢ ³(α₋1)2+λ2´

For large values of λ, the three different terms are equivalent to _λ12, which proves the

absolute integrability of the Fourier transform of f_Mc

c.¤

We can then prove that these functions converges to the call and put option when |M_| tends to in&nity.

4 VOLATILITY SMILE 16

Proposition 15 for i representing either a call or a put, the expectation of the trun-cated payoff converges to the standard payoff

lim |M|→+∞E £ e−rTf_Mi i ¤ =_E£e−rTfi¤

with i=c or p representing either a call or a put option

Proof: Let ε be equal to 1 (call) or ₋1 (put) according to the option type. We have for everyMi ∈Rsatisfying the conditions (17)

Z ¯_¯ e−rTf_Mi _i¯¯dpXT(x) ≤ Z ¯_¯ e−rTfi¯¯dpXT(x) ≤ +_∞ as well as lim |M|→+∞f i Mi =f i

This gives the result by dominated convergence.¤

The same methodology can be applied to the case of a binary option. The trun-cated function can be the following

f_MRange Range(x) =1{ex≥K} ³ 1_{x≤Mc}+e−α(x−Mc)₁ {x>Mc} ´ with α>0.

3.2.3 Methodology for option pricing with the Fourier transform

We have seen that when we know the Fourier exponent of the Levy process as well as the Fourier transform of the truncated payoff, using the proposition 12, we need to multiply these two function and invert their Fourier transform. The proposition 15 shows us that the limit of these truncated payoffoption converges to the standard option. This gives us a methodology for pricing option with the Fourier transform.

4

Volatility Smile

The true motivation of Levy process is to infer some characteristic of our process that take account for the volatility smile. Let us remind the result of static replication (see Carr [1997]), which is used for the static hedging of derivatives product

4 VOLATILITY SMILE 17

Proposition 16 If f belongs to C2(_R+∗) (set of functions twice differentiable with continuous second order derivative function), then for everyx_∈R+, for everyκ_∈R+

f(x) = Z κ 0 (x₋u)+f00(u)du+ Z +∞ κ (u₋x)+f00(u)du

Proof:see Car [1997].¤

4.1

Estimation of the Levy process

Letλ_≤0. If we want to estimate the Laplace exponent, an interesting property is to use the proposition 16, leading to the following proposition:

Proposition 17 Denoting by P ut(u) (respectively Call(u)) the price of a put (re-spectively a call) on ST with strike u, we can calculate the Levy-Laplace exponent as

φ(λ) = 1 T ln µ λ(λ₋1) Sλ 0 Z +∞ 0 erTP ut(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ +σ 2_λ2 2 = 1 T ln µ λ(λ₋1) Sλ 0 Z +∞ 0 erTCall(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ + σ 2_λ2 2

Proof: The function f :x _→xλ belongs to C2(_R+∗). The proposition 16 leads to E · S_Tλ λ(λ₋1) ¸ = Z κ 0 E h (ST −u)+u(λ−2) i du+ Z +∞ κ E h (u₋ST)+u(λ−2) i du

using the two limiting case κ= 0andκ= +_∞, we get

E · S_Tλ λ(λ₋1) ¸ = Z +∞ 0 erTP ut(u)u(λ−2)du = Z +∞ 0 erTCall(u)u(λ−2)du

Using the de&nition of the Levy exponent (equation (4)) , we get the &nal result.¤

4.2

Particular case

In the case of the process given in the section 2.2.2 by equation (5), we get that for the Levy-Laplace exponent the following formula

φ(λ) = θ ³ E ³ (1+U)λ ´ −1 ´ = 1 T ln µ λ(λ₋1) S₀λ Z +∞ 0 erTP ut(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ +σ 2_λ2 2

4 VOLATILITY SMILE 18

which gives us a way of calibrating our model by means of call and put prices of the markets. If we assume furthermore that1+U follows a lognormal law with mean m and volatility v2, we get the closed formula

θ µ eλm+λv 2 2 −1 ¶ = 1 T ln µ λ(λ₋1) Sλ 0 Z +∞ 0 erTP ut(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ +σ 2_λ2 2 = 1 T ln µ λ(λ₋1) S₀λ Z +∞ 0 erTCall(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ + σ 2_λ2 2

The same case provides us as well an explicit formula for the Levy-Khintchine exponent ψ(λ) =iµλ₋σ 2 2 λ 2_+θ µ eimλ−v 2 2λ 2 −1 ¶

we need to estimate the different parameters so that the price of the different options assuming this Levy process is consistent with market data.

4.3

Implication

We have implemented this for different level ofλ, takingσ equal to the implied Black Scholes volatility, µsatisfying the no-arbitrage condition (6)

µ=r₋θ µ em+v 2 2 −1 ¶

If the model above is realistic, we should get that the following function g(λ) = 1 T ln µ λ(λ₋1) Sλ 0 Z +∞ 0 erTCall(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ +σ 2_λ2 2 = 1 T ln µ λ(λ₋1) Sλ 0 Z +∞ 0 erTP ut(u)u(λ−2)du ¶ −λ µ µ₋σ 2 2 ¶ +σ 2_λ2 2

should be equal to a polynomial expression in λsince it is equal toθ µ em+v 2 2 ¶λ −θ.

4.4

Impact on the smile

This leads to the complicated issue of calibrating the model. This is a very compli-cated issue and no simple answer exits. However, we have taken different values of parameters for the Levy process and we have obtained realistic form of smiles. This is summarized by the &gure 1, which shows the evolution of the implied volatility with the time to maturity for the set of parameters θ=1, m=₋0.15,v2_{= 0.20}

5 CONCLUSION 19 22.50% 32.50% 42.50% 52.50% 62.50% 72.50% -2 -1 0 1 2 6 month 9 month 12 month 15 mpnth 18 month 21 month 24 month

Figure 1: Volatility Smile implied by a Levy process with jumps modelled as a Poisson process with a lognormal intensity

5

Conclusion

In this paper, we have seen that the use of Levy processes enables us to take account for the volatility smile. The approach adopted here is a semi-parametric one based on a Levy process description of our economy. This has the great advantage to encompass many previous works since Levy process includes Brownian motion as well as many jump processes. We show that the Fourier transform can lead to an efficient way of getting prices after inferring the Levy-Khintchine exponent.

There are many possible extensions to this work. The &rst one concerns some empirical studies to quantify the &t of this model with market data. This is a com-plicated issue like all calibration procedure and test of goodness of &t. The second one is to develop more efficient numerical procedure based on Fast Fourier Transform algorithm that take account of the particular situation developed here.

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